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Department
Mathematics
Course
Mathematics 1229A/B
Professor
Prof
Semester
Fall

Description
Math1229ABUnit1VectorstextreferenceSection11cVOlds2010Unit111VectorsYouarefamiliarwiththesetofrealnumbersRealnumbersmeansallthenumbersyouveeverheardoforcanimagineunlessyouvelearntaboutoratleastheardofimaginarynumberstheyrenotrealnumbersRealnumbersincludesalltheintegersincludingbothpositiveandnegativeandofcourse0fractionscalledrationalnumbersdecimalsthatcantbeexpressedasfractionstheirrationalnumbersAllthenumbersalongtherealnumberlinefromtoWecallthesetofrealnumbersYourealsofamiliarwiththexyplaneYouvedrawngraphsinthisplaneTheresthexaxishorizontalandtheyaxisverticalwhichcrossattheoriginEachaxisisbasicallyarealnumberlineAnypointinthisplanecanbeexpressedasanorderedpairxygivingitsxcoordinateanditsycoordinateEachcoordinatecanbeanyrealnumberWecallthexyplane2spaceandthe2setcontainingallofthepointsinthisplaneiscalledWepronouncethatR2ieArtoo2SocanbethoughtofasthesetofallorderedpairsofrealnumbersYouveprobablyalsoseensomethingevenmorecomplicatedwith3axesthexaxistheyaxisandthezaxisEachaxisisperpendiculartobothoftheotherswhichmakesithardtodrawonapieceofpaperorablackboardTheyreoftendrawnwiththeyaxishorizontalthezaxisverticalandthexaxisoatafunnyangletorepresentthatitscomingstraightoutofthepageatyouThe3axesrepresentthe3dimensionsofspaceierealityLiketheroomyouresittinginTheresnotonlyupanddownandleftandrightbutalsonearandfarorhereandthereorwellyouknowthatthirddimensionwhichwemightcalldepthInMathwecalltheregiondenedbythese3axes3spaceAndpointsin3spacearerepresentedbyorderedtriplesxyzgivingthexyandzcoordinatesofthepointAsbeforeeachofthesecoordinatescanbeanyrealnumberThe3setcontainingallofthepointsin3spaceiscalledpronouncedR3andwecanthinkofthisasthesetofallorderedtriplesofrealnumbersNowthatyouvegotthatstraightletsconfusethingsSometimeswhenwewriteanorderedpairoranorderedtripleitdoesntrepresentapointInsteaditrepresentsadirectedlinesegmentcalledavectorAndwhenthepairortriplerepresentsavectorthenumbersinitorsymbolsrepresentingnumbersarentcalledcoordinatestheyrecalledcomponentsThatseemslikeitwillbeconfusingusingthesamenotationtodenotetwodierentthingsItsnottoobadthoughbecauseyoucantellbythecontextwhetherapairortripletisapointoravectorAndbecausewewritethenamesdierentlydependingwhetheritsapointoravectorPointsarenamedwithnormalcapitallettersusuallyPorsomethingnearbyinthealphabetandsometimeswithsubscriptsSowemighthavethepointsPppandQqqorthepoints1212PxyzandPxyzWithvectorswedontusecapitallettersandwedosomething11112222toshowthatitsavectorInourtextbooktheyuseboldfacetypeforthenameofavectorForinstancethevectorvvvorthevectoruuuuSureIhearyouthinkingthats12123easyenoughforyoubutwhataboutmewhenImwritingwithapenorpencilWellthatswhyweregoingtouseadierentconventioninthesenotesOnethatsmuchmoreobviousWhenaletteristhenameofavectorwellputanarrowoveritLikethisvvv12SowhyisitthatwerepresentavectorwhichwesaidisadirectedlinesegmentusingsomethingthatlookslikeapointWellitsjustaconventionItsshorthandWhenwesayv12whatwemeanisthatthevectorvisthedirectedlinesegmentthatstartsattheoriginandendsatthepoint12Soheresapictureofthevectorv12Itstartsatthepoint00andthenendsataplacethats1unittotherightand2unitsupfromtherey632v10123x
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